Semester Exam 2 – Phys4312 – Fall 2021
This is a take-home exam. The exam is due on Monday, (11/22) at 11:00am in class. Under no
circumstances the exam will be accepted after the deadline. As you take the exam, you are not
allowed to communicate with anyone but me, and you are only allowed to consult your class
notes and the McIntyres's Quantum Mechanics textbook.
1 (35 points) – A beam of identical neutral particles with spin ½ is prepared in the |+> state. The beam
enters a uniform magnetic field ?0 , which is in the xz-plane and makes an angle θ with the z-axis. After
a time T in the field, the beam enters a Stern-Gerlach analyzer oriented in the y-axis. Show that the
probability that the particles will be measured to have spin up in the y-direction is
?+? =
1
2
(1 − sin ? sin(?0?))
2 (35 points) – A quantum mechanical system has the initial state |?(? = 0) > = 3 5⁄ |?1 > +
4
5⁄ |?2
where |?? > are the normalized eigenstates of the operator �̂� co
esponding to the eigenvalues ??. In
this |?? > basis, the Hamiltonian of the system is represented by the matrix
? = ?0 (
2 1
1 2
)
a) If you measure the energy of this system, what values are possible, and what are the
probabilities of measuring those values?
) Show that the expectation value of �̂� is
?1+?2
2
+
7(?2−?1)
50
cos(
2?0
ħ
?)
3 (30 points) – A particle in an infinite square well potential has an initial state vector
|?(? = 0) > =
1
√5
(|?1 > −2?|?2 >) where |?? > are the energy eigenstates of the Hamiltonian
operator. Show that the expectation value of the position is
? >= ? (
1
2
+
64
45?2
sin (
3?2ħ
2??2
?))
Useful integrals:
∫ ? ????(??)?? =
??
?
−
???(???)
???
−
? ???(???)
??
∫ ? ???(??)???(???)?? =
????(??) − ???(???) + ???? ????(??)
????